3 Answers
3

It is not that messy the root test. As $\lim a_n/n^3 = 1$ you have for sufficiently large $n$ that
$1/2 \le a_n/n^3 \le 3/2$ so $n^3/2 \le a_n \le 3n^3/2$.
And the $n$-th root of both sides of the inequality is seen easily to approach $1$.